protein dynamics, allostery and function · 2017. 7. 5. · nuclear magnetic resonance...
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Protein Dynamics, Allostery and Function
Lecture 3. Protein Dynamics
Xiaolin Cheng UT/ORNL Center for Molecular Biophysics
SJTU Summer School 2017 1
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Obtaining Dynamic Information
• Experimental Approaches – X-‐ray Crystallography – Nuclear MagneBc Resonance – Neutron Spectroscopy – H/D exchange mass spectrometry – Single-‐Molecule Fluorescence Spectroscopy
• ComputaBonal Approaches – Molecular Dynamics – Normal Mode Analysis – ElasBc Network Model – Generalized Langevin EquaBon
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X-ray Crystallography
• B-‐factors indicate the amount of thermal fluctuaBon for a parBcular residue. Is anisotropic but usually only measurable as an isotropic mean value.
• MoBon extrapolated between crystal structures of two different conformaBonal states.
• Weakness: ConformaBonal changes are not necessary linear.
Maltose Binding Protein Engulfing Ligand
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Nuclear Magnetic Resonance
Solution NMR techniques cover the complete range of dynamic events in enzymes.
Chem. Rev. 2006, 106, 3055-3079
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Nuclear Magnetic Resonance Fast Backbone and Side-Chain Motions
Occur on the ps to ns timescale, usually involving measuring relaxation times such as T1 and T2 to determine order parameters, correlation times, and chemical exchange rates.
where τm is the correlation time for the overall rotational diffusion of the macromolecule, S2 is the order parameter, and 1/τ ) 1/τm + 1/τe, where τe is the time scale (ns) for the internal bond vector (e.g. N-H) motions.
The spectral density function, J(ω), is directly related to the three relaxation rates, through the Lipari and Szabo model,
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Nuclear Magnetic Resonance Conformational Exchange
Residual dipolar coupling (RDC) provides spatially and temporally averaged information about an angle between the external magnetic field and a bond vector in a molecule
RDCs also provide rich geometrical information about dynamics on a slow (>10−9 s) in proteins. A motion tensor B can be computed from RDCs:
The RDC-‐derived moBon parameters are local measurements.
The dipolar coupling between two nuclei depends on the distance between them, and the angle of bond relative to the external magnetic field.
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7 QENS / MD
Neutron Scattering
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8 QENS / MD
Dynamic Structure Factor
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9 QENS / MD
Quasi-elastic Neutron Scattering Simple Translational Diffusion
In the energy space, we typically fit with Lorentzians.
For simple translational diffusion, particles follow Fick’s Law:
Γ is the half width at half maximum (HWHM) which is this case = DQ2, leading to a broadening of the elastic line following a given Q-dependence.
Ds is diffusion coefficient
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10 QENS / MD
Neutron Spin Echo
Neutron Spin Echo (NSE) measures collective dynamics at the timescale of ns – 100 ns. It provides the intermediate scattering function, I(Q, t), directly.
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11 QENS / MD
Neutrons are unique for probing protein H dynamics – both spatial and time resolution.
Inelastic Neutron Scattering
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Hydrogen/Deuterium Exchange Mass Spectrometry
Amide hydrogens exchange at rates that are characteristic of local backbone conformation and dynamics. In highly dynamic unstructured regions, the exchange reaction proceeds on the msec-sec timescale while amides that are hydrogen bonded will exchange more slowly (minutes to days).
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Single-Molecule Fluorescence Spectroscopy
Förster resonance energy transfer (FRET), fluorescence resonance energy transfer (FRET), resonance energy transfer (RET) or electronic energy transfer (EET) is a mechanism describing energy transfer between two light-sensitive molecules (chromophores).
In FRET, the efficiency of the absorption of the photon emitted from the first probe in the second probe depends on the distance between these probes. Since the distance changes with time, this experiment probes the internal dynamics of the molecule.
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Obtaining Dynamic Information
• Experimental Approaches – X-‐ray Crystallography – Nuclear MagneBc Resonance – Neutron Spectroscopy – H/D exchange mass spectrometry – Single-‐Molecule Fluorescence Spectroscopy
• ComputaBonal Approaches – Molecular Dynamics – Normal Mode Analysis – ElasBc Network Model – Generalized Langevin EquaBon
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In Silico Approaches
Different representation of protein structures and dynamics
Molecular Dynamic Simulation Normal Mode Analysis Elastic Network Analysis Generalized Langevin Equation
Accuracy
Detail
Com
putation Time
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Molecular Dynamics Simulation
),...,xV(xxm nii 1−∇=!!
∑∑∑∑
∑∑∑
>> ⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛+
⋅+
−++−+−=
i ij ij
ij
ij
ijij
i ij ij
ji
diheanglebondb
rrrqq
nKKbbKV
612
02
02
0
4
))cos(1()()(
σσε
φφθθ φθ
Newton’s Equation of Motion
Energy Function (Force Field)
The ultimate detail modeling of protein fluctuation, every atom is accounted for. How the system evolves with time - computational microscope enzyme functional dynamics, receptor gating mechanism…
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MD in the 1970s and Now
1977 - Bovine Pancreatic Trypsin Inhibitor ~400 atoms in vacuum, for 9 ps
Model of the cytoplasm of B. subtilis 60 million atoms in water, for >1µs
~1015
McCammon, Gelin, Karplus. Nature, 1977, 267, 585
1979 - Simulations resulted in the recognition that B factors can be used to infer internal motions.
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Simulated Vibrational Spectra
Which Regions Contribute to the Observed Differences in the INS?
POX
Pitch
Amplitu
de
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t = 100 ps ×5, 10 ns ×5, 500 ns ×5 and 17 µs
Protein Conformational State
PGK: 45 kD
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Markov State Model
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Markov State Model
What can we know from an MSM?
• Thermodynamic information – Equilibrium populations
• Kinetic information – Mean first passage time – Dominant Pathways (pathways with major flux)
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Normal Mode Analysis
A simple harmonic oscillator based system that analyzes dynamics near a local minimum. Specifically, if one expands the potential energy function U around a minimum on the energy surface, ro, the Hamiltonian of the system is given by:
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Normal Mode Analysis
Harmonic oscillator
Newton law:
ma = F
ma = −kx
x = Acos(ωt + φ)
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Normal Mode Analysis
coordinate transformation
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Normal Mode Analysis
normal mode coordinate
normal mode vector
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Normal Mode Analysis
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Normal Mode Analysis
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Normal Mode Analysis
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Normal Mode Analysis Atomic fluctuations
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Normal Mode Analysis Cartesian displacement along normal mode
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Normal Mode Analysis Overlap
Gives information on how well a normal mode describe the conformational change. A value close to 1 indicates that the mode represents well the conformational change.
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Normal Mode Analysis Correlation of atomic motion
= 1 => atom i and j moves in the same direction with same amplitude: correlated motions = -1 => atom i and j moves in the opposite direction with same amplitude: anti-correlated motions = 0 => no correlation
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Quasi-harmonic Analysis
∑=Δi i
ijBj Tkx 2
22 ||
ω
α
ijBji FTkxx )( 1−=ΔΔ )/( 2jiij xxVF ∂∂∂≡
Cartesian atomic fluctuations can be determined as a sum over all normal modes
Under quasi-harmonic approximation, the sum can be simplified to yield
ji xxH ΔΔ=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
2
21
000...0
00
n
B
B
Tk
Tk
L
ω
ω
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Elastic Network Model Atoms connected via elastic springs
Coarse grained models such as only Cα atoms can be used
Tirion MM (1996) Phys Rev Lett. 77, 1905-1908
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Elastic Network Model
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Elastic Network Model
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Brownian Motion 1827: Robert Brown observed pollen particles moving erratically in water in a zigzag fashion 1905: Albert Einstein and Marian von Smoluchowski explained the phenomenon mathematically 1911: Jean Perrin validated the theory by the use of sedimentation equilibrium
http://www.deutsches-museum.de/ausstell/dauer/physik/e_brown.htm
Small particles, moving with thermal motion, randomly collide with large particle à diffusive, random motion of large particle in a fluid
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Brownian Motion 1827: Robert Brown observed pollen particles moving erratically in water in a zigzag fashion 1905: Albert Einstein and Marian von Smoluchowski explained the phenomenon mathematically 1911: Jean Perrin validated the theory by the use of sedimentation equilibrium
For n-D random walk:
,22 nDtx = ζπη /6/ TkrTkD BB ==
D = diffusion coefficient T = temperature r = hydrodynamic radius η = solvent viscosity ζ = friction coefficient
Stokes-Einstein relation for Brownian motion
Average displacement of a particle is proportional to square root of time
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Brownian Motion in a Living Cell
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Modelling Brownian Motions Discrete: stochastic motions of individual particles • Monte Carlo • Langevin • Brownian dynamics
Continuum: concentration distribution probability
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Langevin Equation
)()()0(,0)( tm
tSStS δγ
==!!!
DttmTkRtRR B 66))((,0 2
0 ==−=γ
!!!
,)()(2
2
SFdttRd
dttRdm
!!!!
++−= γ
Friction: force in opposite direction and proportional to velocity : systematic interaction forces : stochastic forces γ : friction coefficient = kBT/D
F!
S!
SFdttRd !!!
γγ11)(
+=
Under over-dampened conditions, leading to Brownian dynamics
Brownian motion of a particle in a fluid due to collisions with the molecules of the fluid can be described by the Langevin equation,
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Ermak-McCammon Brownian Dynamics
tDRRR ijjii Δ== 2,0NitRtFDTk
trttr ij
jijB
ii ...,,1),(1)()( =Δ+Δ+=Δ+ ∑ri: coordinates of particle i Fj: systematic interaction forces Dij: diffusion tensor à hydrodynamic interactions Ri: random stochastic displacement
TkDmt Biii />>Δtime step >> momentum relaxation time Ermak and McCammon JCP(1978) 69,1352-60
Hydrodynamic interaction (HI) between solute molecules results from induced fluid flow of solvent due to motion of solute molecules
tDRR ijji Δ= 2 Cholosky decomposition
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Molecular Diffusion
The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. When applied to particle position distributions, it becomes the Smoluchowski equation. When zero diffusion, it is known as the Liouville equation.
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Smoluchowski Equation
The Smoluchowski equation is the Fokker–Planck equation for the probability density function of the particle positions of Brownian particles.
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)](1[ 2 PWTk
PDtP
B
⋅∇∇−∇=∂∂
Continuity equation
Smoluchowski Equation
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)](1[ 2 PWTk
PDtP
B
⋅∇∇−∇=∂∂
Diffusive association of particles to a sphere
0/ =∂∂ tcsteady state: 02
2 )(12 ==∇drrcd
rc
( )racrc −= ∞ 1)(after integrating:
particle flux: 2)(raDc
drdcDrJ ∞−=−=
number of collisions per second at r = a: aDcaaJaI ∞−== ππ 44)()( 2
association rate: DacaIka π4/)( =−= ∞ ~ 109 M-1s-1
cDtc 2∇=∂∂Diffusion equation:
(without friction)
Smoluchowski Equation
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Fokker-Planck Equation
a partial differential equation that describes the time evolution of the probability density function of the momentum of a particle undergoing a Brownian motion.
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a partial differential equation that describes the time evolution of the probability density function of the momentum of a particle undergoing a Brownian motion.
The equation can be generalized to other observables:
Fokker-Planck Equation